C1-algebras without idempotents

The simplest statement of the main results are these: Let π be a free group on 2 generators. Let Cπ be the complex ring and L¹π the ring extension to L¹ sums. Then L¹π contains no idempotents. Furthermore, if α ? Cπ, β?L¹π are nonzero, then αβ ≠ 0. The first result is in the direction of proving that a certain simple $\text{C}{}^1\text{-}\text{algebra}$ has no idempotents yielding a counter-example to the suggestion that simple $\text{C}{}^1\text{-}\text{algebras}$ are generated by their projections.